All practice questions

Question 1Question →

Many urban planners argue that expanding municipal light rail systems inevitably reduces traffic congestion in suburban corridors. However, a recent comprehensive study revealed that suburban commuters rarely shift from private automobiles to light rail unless parking fees in urban business districts increase significantly. Consequently, proposing light rail expansion without simultaneous adjustments to municipal parking pricing will fail to relieve suburban traffic. Implementing higher parking fees will incentivize a sufficient number of commuters to adopt public transit, thereby achieving the desired reduction in highway congestion.

In the argument above, what are the respective roles played by the two boldfaced portions?

The first is evidence cited to challenge a position that the argument opposes; the second provides a rationale for a key condition necessary for the proposed solution to succeed.

The first is the main conclusion of the argument; the second is a premise offered in support of that conclusion.

The first is a counterargument directed against the author's primary recommendation; the second is the author's main conclusion.

The first provides background information establishing the context of transit planning; the second is an intermediate conclusion that directly refutes the urban planners' claim.

The first is evidence supporting the traditional view held by urban planners; the second is a concession made to those who oppose municipal transit funding.

Show answer & explanation

Answer: The first is evidence cited to challenge a position that the argument opposes; the second provides a rationale for a key condition necessary for the proposed solution to succeed.

Answer

The option stating that the first bolded portion is evidence cited to challenge a position that the argument opposes, and the second provides a rationale for a key condition necessary for the proposed solution to succeed.

The correct option accurately evaluates both bolded statements: the first bolded statement presents empirical study findings that counter the claim of urban planners (a position the author opposes), while the second bolded statement explains the mechanism by which higher parking fees achieve reduced highway congestion, functioning as a supporting rationale for the essential condition identified by the author.

Step-by-Step Solution

1

Analyze the overall argument structure and identify the author's main conclusion.

The argument begins with an opposing view ('Many urban planners argue...'). The author's main conclusion is that proposing light rail expansion without adjusting parking fees will fail to relieve traffic.

Establishing the main conclusion enables accurate classification of surrounding supporting statements.

2

Determine the functional role of the first bolded statement.

The first bolded statement presents study findings demonstrating that commuters do not switch to light rail unless parking fees rise. This evidence directly undermines the urban planners' traditional view and supports the author's conclusion.

The contrastive transition 'However' signals that the study evidence refutes the initial position.

3

Determine the functional role of the second bolded statement.

The second bolded statement explains why raising parking fees works—by incentivizing commuters to adopt public transit—thereby justifying the condition required for transit success.

It provides the underlying mechanism/rationale for why parking fee adjustments are essential.

4

Match the structural analysis to the correct option.

The option identifying the first statement as evidence challenging an opposed position and the second statement as a rationale for a key condition is correct.

It captures the exact logical relationship of both bolded statements to the argument.

Key Concept

Analyzing Bolded Statement Roles in Critical Reasoning

Question 2Question →

If $x$ is a nonzero real number, is $\frac{x^2 - 4x}{x} > 0$ ?

(1) $x > 5$
(2) $x^2 > 16$

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

Answer: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Simplifying the target expression

\frac{x^2 - 4x}{x}

for

x \neq 0

yields

x - 4

. Thus, the question target rephrases to 'Is

x > 4

?'. Statement (1) states

x > 5

, which guarantees

x > 4

, yielding a definitive 'Yes' answer. Statement (2) allows

x > 4

or

x < -4

, yielding both 'Yes' and 'No' outcomes. Therefore, Statement (1) alone is sufficient, but Statement (2) alone is not sufficient.

Step-by-Step Solution

1

Simplify the question stem expression

Since

x \neq 0

, factor out

x

from the numerator:

\frac{x(x - 4)}{x} = x - 4

.

Simplifying the expression reveals the underlying condition required by the question stem.

2

Rephrase the target question

The original question 'Is

\frac{x^2 - 4x}{x} > 0

?' simplifies to 'Is

x - 4 > 0

?', which is equivalent to 'Is

x > 4

?'.

Target rephrasing turns a fraction inequality into a simple comparison.

3

Evaluate Statement (1):

x > 5

If

x > 5

, then

x

is strictly greater than 4. The answer to 'Is

x > 4

?' is a definitive YES.

Since Statement (1) provides a conclusive 'Yes', Statement (1) alone is sufficient.

4

Evaluate Statement (2):

x^2 > 16

Taking the square root gives

|x| > 4

, meaning

x > 4

or

x < -4

.
- If

x = 5

, then

x > 4

(YES).
- If

x = -5

, then

x < 4

(NO).

Because Statement (2) allows both 'Yes' and 'No' answers, it is not sufficient.

Key Concept

Question Stem Simplification and Target Rephrasing

Question 3Question →

Is $x$ an integer?

(1) $3x$ is an integer.
(2) $5x$ is an integer.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

Answer: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Answer

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Evaluating Statement (1) alone shows

x

could be

\frac{1}{3}

(not an integer) or

1

(an integer), so Statement (1) is insufficient. Evaluating Statement (2) alone shows

x

could be

\frac{1}{5}

(not an integer) or

1

(an integer), so Statement (2) is insufficient. Taking both statements together,

3x = a

and

5x = b

for integers

a

and

b

. Subtracting

5x

from

2(3x)

yields

6x - 5x = x = 2a - b

. Because integers are closed under multiplication and subtraction,

2a - b

must be an integer, confirming that

x

is definitively an integer.

Step-by-Step Solution

1

Evaluate Statement (1) independently

If

3x = a

where

a

is an integer, then

x = \frac{a}{3}

. If

a = 1

,

x = \frac{1}{3}

(not an integer). If

a = 3

,

x = 1

(an integer).

Since

x

can be either an integer or a non-integer, Statement (1) alone is not sufficient.

2

Evaluate Statement (2) independently

If

5x = b

where

b

is an integer, then

x = \frac{b}{5}

. If

b = 1

,

x = \frac{1}{5}

(not an integer). If

b = 5

,

x = 1

(an integer).

Since

x

can be either an integer or a non-integer, Statement (2) alone is not sufficient.

3

Evaluate Statements (1) and (2) together

Since

3x

and

5x

are both integers, their linear combination

2(3x) - 5x = 6x - 5x = x

must also be an integer.

The difference between two integers is always an integer, proving conclusively that

x

is an integer.

Key Concept

Linear combinations of real numbers and integer constraints in Data Sufficiency

Estimated Time:1m 0s

Question 4Question →

Over the past year, a gourmet coffee subscription company recorded a significant increase in the total number of monthly account cancellations. However, during that same year, the company's monthly cancellation rate—defined as the percentage of total active subscribers who canceled their accounts each month—steadily decreased.

Which of the following, if true, best helps to resolve the apparent paradox described above?

The total number of active subscribers to the coffee service increased substantially over the course of the year.

Customer satisfaction surveys showed that most subscribers who canceled were dissatisfied with recent delivery delays.

The subscription company increased its monthly service fees substantially at the start of the year.

The company introduced several new premium coffee roasts mid-way through the year.

Competing subscription services experienced a similar rise in the overall number of customer cancellations.

Show answer & explanation

Answer: The total number of active subscribers to the coffee service increased substantially over the course of the year.

Answer

The total number of active subscribers to the coffee service increased substantially over the course of the year.

The correct answer resolves the paradox by identifying a shift in the overall subscriber base. Total cancellations equal the cancellation rate multiplied by the total number of active subscribers. If the total subscriber base grew substantially, a lower percentage cancellation rate would still yield a higher absolute number of cancellations.

Step-by-Step Solution

1

Identify the two contradictory facts in the stimulus.

Fact 1: Total monthly cancellations increased. Fact 2: The monthly cancellation rate (percentage of active subscribers canceling) decreased.

Resolving a paradox requires finding an underlying factor that allows both facts to exist simultaneously without denying either premise.

2

Analyze the relationship between rate and total volume.

Total Cancellations = (Total Active Subscribers) × (Cancellation Rate).

If the cancellation rate drops while the total number of cancellations increases, the total number of active subscribers must have expanded.

3

Evaluate the option choices to find the statement establishing subscriber base growth.

The option stating that active subscribers increased substantially provides the missing mathematical linkage.

A much larger denominator (total active subscribers) yields a higher total count of cancellations even when the percentage fraction is smaller.

Key Concept

Resolving Rate versus Total Volume Discrepancies

Question 5Question →

Read the following sentence regarding historical forest ecology:

"While early twentieth-century silviculturists asserted that the rapid decline of old-growth oak stands in the Baltic basin was driven primarily by overharvesting for naval shipbuilding, recent dendrochronological analysis reveals that severe decadal drought cycles in the late seventeenth century had already compromised root systems, rendering these stands uniquely susceptible to fungal pathogens prior to the onset of intensive timber extraction."

Which of the following inferences regarding the decline of Baltic oak stands is most strongly supported by the statement above?

The vulnerability of Baltic oak stands to fungal pathogens was heightened by environmental stress experienced prior to the expansion of intensive timber harvesting.

Intensive timber harvesting for naval construction posed a greater long-term ecological threat to Scandinavian forests than to Baltic basin oak stands.

Late seventeenth-century dendrochronological records confirm that naval shipbuilding was the primary driver of oak depletion, validating early silvicultural theories.

Twentieth-century silviculturists maintained that fungal pathogens, rather than drought, were responsible for compromising the root structures of Baltic oaks.

Timber extraction had absolutely no impact on the overall decline of Baltic oak stands compared to the devastating effects of fungal pathogens.

Show answer & explanation

Answer: The vulnerability of Baltic oak stands to fungal pathogens was heightened by environmental stress experienced prior to the expansion of intensive timber harvesting.

Answer

The vulnerability of Baltic oak stands to fungal pathogens was heightened by environmental stress experienced prior to the expansion of intensive timber harvesting.

The sentence specifies that drought cycles in the late seventeenth century compromised root systems and rendered oak stands uniquely susceptible to fungal pathogens prior to the onset of intensive timber extraction. This directly supports the conclusion that environmental stress occurring before expanded harvesting heightened the trees' vulnerability to pathogens.

Step-by-Step Solution

1

Analyze the logical structure of the target sentence.

The sentence contrasts an older view (silviculturists claiming shipbuilding overharvesting was the main cause) with recent dendrochronological evidence.

Identifying the main clause clarifies what the author actually asserts as factual.

2

Examine the specific temporal and causal sequence presented in the evidence.

Late 17th-century drought cycles -> compromised root systems -> made trees susceptible to pathogens -> occurred prior to intensive timber extraction.

Evaluating pre-existing conditions confirms how drought stress preceded and enabled fungal vulnerability.

3

Match the sequence to the correct inference.

The statement directly supports that pre-harvesting environmental stress (drought) increased susceptibility (vulnerability) to pathogens.

A valid GMAT single-sentence inference must rely strictly on stated facts without external assumptions.

Key Concept

Single-Sentence Logical Inference

Question 6Question →

A marine conservation agency abides by the following guidelines regarding commercial oceanic operations: An oceanic research institute will approve a coastal zone for deep-sea mining only if the zone undergoes an ecological impact assessment. An ecological impact assessment is conducted whenever a designated marine habitat contains endangered coral species, unless the habitat is classified as a protected sanctuary. Furthermore, any coastal zone that undergoes an ecological impact assessment requires the implementation of continuous acoustic monitoring.

If the statements above are true, which of the following MUST also be true?

A coastal zone cannot be approved for deep-sea mining without the implementation of continuous acoustic monitoring.

Any coastal zone in which continuous acoustic monitoring is implemented will automatically be approved for deep-sea mining.

If a designated marine habitat is classified as a protected sanctuary, an ecological impact assessment cannot be conducted under any circumstances.

Any designated marine habitat that contains endangered coral species must implement continuous acoustic monitoring.

A coastal zone that does not contain endangered coral species is not required to implement continuous acoustic monitoring under any circumstances.

Show answer & explanation

Answer: A coastal zone cannot be approved for deep-sea mining without the implementation of continuous acoustic monitoring.

Answer

A coastal zone cannot be approved for deep-sea mining without the implementation of continuous acoustic monitoring.

The passage establishes a chain of necessary conditions: approval for deep-sea mining requires an ecological impact assessment (Approval -> Assessment), and an ecological impact assessment requires continuous acoustic monitoring (Assessment -> Monitoring). Combining these transitive statements yields Approval -> Monitoring. The contrapositive of this deduction is that without continuous acoustic monitoring, a zone cannot receive approval for deep-sea mining.

Step-by-Step Solution

1

Formalize the conditional statements provided in the passage.

Statement 1: Approval -> Assessment. Statement 2: Coral AND NOT Sanctuary -> Assessment. Statement 3: Assessment -> Monitoring.

Mapping the premises into logical notation clarifies valid deductions and prevents common fallacies.

2

Chain the conditional statements linking Approval, Assessment, and Monitoring.

Approval -> Assessment -> Monitoring. Thus, Approval -> Monitoring.

Transitive logic allows combining two nested conditional rules sharing a middle term (Assessment).

3

Take the contrapositive of the combined deduction.

NOT Monitoring -> NOT Approval.

A conditional statement is logically equivalent to its contrapositive, confirming that without monitoring, approval is impossible.

Key Concept

Conditional Logic and Formal Deductions

Question 7Question →

Under municipal transit regulations, a technology firm receives a commercial testing permit for autonomous trucks on public highways only if its navigation software achieves a Tier-1 safety certification. Achieving a Tier-1 safety certification requires completing at least 500,000 miles of simulated urban driving without a single critical system disengagement. Furthermore, unless a firm holds a commercial testing permit, it is forbidden from deploying autonomous vehicles during peak traffic hours.

If the statements above are true, which of the following must also be true?

A technology firm that deploys autonomous vehicles on public highways during peak traffic hours has completed at least 500,000 miles of simulated urban driving without a critical system disengagement.

Any technology firm that completes 500,000 miles of simulated urban driving without a critical system disengagement will automatically receive a commercial testing permit.

A technology firm that has not been permitted to deploy autonomous vehicles during peak traffic hours has failed to achieve a Tier-1 safety certification.

Autonomous trucks that have logged 500,000 miles of simulated urban driving are safer than conventional commercial trucks operated by human drivers.

Technology firms holding a commercial testing permit deploy autonomous trucks exclusively during non-peak traffic hours.

Show answer & explanation

Answer: A technology firm that deploys autonomous vehicles on public highways during peak traffic hours has completed at least 500,000 miles of simulated urban driving without a critical system disengagement.

Answer

A technology firm that deploys autonomous vehicles on public highways during peak traffic hours has completed at least 500,000 miles of simulated urban driving without a critical system disengagement.

The passage establishes a chain of necessary conditions: to deploy during peak traffic hours, a firm must hold a commercial testing permit; to have that permit, it must hold Tier-1 certification; and to achieve Tier-1 certification, it must complete 500,000 simulated miles without a critical disengagement. Therefore, any firm deploying vehicles during peak hours must have completed those 500,000 simulated miles.

Step-by-Step Solution

1

Formalize the conditional statements given in the passage

Statement 1: Commercial Permit -> Tier-1 Certification. Statement 2: Tier-1 Certification -> 500,000 Simulated Miles without critical disengagement. Statement 3: Deploy Peak Hours -> Commercial Permit.

Translating natural language indicators ('only if', 'requires', 'unless') into formal logic clarifies necessary and sufficient conditions.

2

Chain the conditional statements together transitively

Deploy Peak Hours -> Commercial Permit -> Tier-1 Certification -> 500,000 Simulated Miles.

Because each condition is necessary for the preceding one, the entire chain must hold true.

3

Deduce the logically necessary conclusion

Deploying during peak traffic hours necessarily implies having completed at least 500,000 miles of simulated driving without a critical disengagement.

If the antecedent of the chain is true, all necessary conditions down the chain must also be true.

Key Concept

Transitive Chaining of Necessary Conditions

Question 8Question →

A bag contains 5 red marbles and 5 blue marbles. If 2 marbles are randomly selected from the bag one after another without replacement, what is the probability that at least one of the selected marbles is red?

\frac{7}{9}

\frac{2}{9}

\frac{5}{9}

\frac{3}{4}

\frac{1}{4}

Show answer & explanation

Answer: $\frac{7}{9}$

Answer

\frac{7}{9}

To find the probability of drawing at least one red marble, calculate the probability of the complementary event (drawing two blue marbles) and subtract it from 1. The probability of selecting a blue marble on the first draw is

\frac{5}{10}

, and on the second draw without replacement it is

\frac{4}{9}

. The probability of both marbles being blue is

\frac{5}{10} \times \frac{4}{9} = \frac{2}{9}

. Subtracting this from 1 gives

1 - \frac{2}{9} = \frac{7}{9}

.

Step-by-Step Solution

1

Identify the complementary outcome

The event 'at least one marble is red' is complementary to the event 'no marbles are red' (i.e., 'both marbles are blue').

Using the relationship

P(\text{at least one red}) = 1 - P(\text{both blue})

is simpler than summing individual favorable cases.

2

Calculate the probability of drawing two blue marbles without replacement

The probability that the first marble is blue is

\frac{5}{10} = \frac{1}{2}

. The probability that the second marble is blue is

\frac{4}{9}

. Therefore,

P(\text{both blue}) = \frac{5}{10} \times \frac{4}{9} = \frac{20}{90} = \frac{2}{9}

.

Since draws are without replacement, the sample space and number of blue marbles decrease by 1 after the first draw.

3

Subtract the complementary probability from 1

P(\text{at least one red}) = 1 - \frac{2}{9} = \frac{7}{9}

.

The sum of complementary probabilities is always equal to 1.

Key Concept

Complementary Probability:

P(\text{at least one}) = 1 - P(\text{none})

Question 9Question →

If $x$ is a positive real number, is $x$ an integer?

(1) $x^3$ is an integer.
(2) $x^5$ is an integer.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

Answer: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Answer

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Evaluating each statement independently leaves open the possibility that

x

is an irrational root like

\sqrt[3]{2}

or

\sqrt[5]{2}

, making neither statement alone sufficient. Combining both statements allows us to express

x

as

\frac{(x^3)^2}{x^5}

. Because both

x^3

and

x^5

are integers,

x

must be a rational number. For any rational number whose cube is an integer, its denominator must equal 1, which proves conclusively that

x

is an integer. Thus, both statements together are sufficient.

Step-by-Step Solution

1

Evaluate Statement (1) independently.

Statement (1) is insufficient.

If

x = 2

, then

x^3 = 8

(an integer), so

x

is an integer (Yes). However, if

x = \sqrt[3]{2}

, then

x^3 = 2

(an integer), but

x

is not an integer (No). Since both Yes and No answers are possible, Statement (1) alone is not sufficient.

2

Evaluate Statement (2) independently.

Statement (2) is insufficient.

If

x = 2

, then

x^5 = 32

(an integer), so

x

is an integer (Yes). However, if

x = \sqrt[5]{2}

, then

x^5 = 2

(an integer), but

x

is not an integer (No). Since both Yes and No answers are possible, Statement (2) alone is not sufficient.

3

Combine Statement (1) and Statement (2).

The combined statements are sufficient.

Notice that

\frac{(x^3)^2}{x^5} = \frac{x^6}{x^5} = x

. Since

x^3

is an integer,

(x^3)^2

is also an integer. Since

x^5

is an integer,

x = \frac{(x^3)^2}{x^5}

is the ratio of two integers, meaning

x

must be a rational number. If

x = \frac{a}{b}

in simplest form where

a

and

b

are positive integers with

\gcd(a,b)=1

, then

x^3 = \frac{a^3}{b^3}

being an integer implies

b^3 = 1

, so

b = 1

. Thus,

x = a

, which means

x

must be an integer. This yields a definitive 'Yes' answer.

Key Concept

Integer constraints versus real numbers in Data Sufficiency and quotient relationships of exponent powers

Question 10Question →

To combat declining municipal revenues from downtown parking meters, the city council of Elmridge plans to eliminate the two-hour parking limit on all downtown streets. Council members argue that allowing drivers to park indefinitely will encourage visitors to spend more time dining and shopping in the downtown district, thereby increasing local business sales and boosting the city's sales tax revenues.

Which of the following, if true, most seriously weakens the city council's argument?

Eliminating time limits causes downtown office commuters to occupy parking spaces early in the morning for full workdays, drastically reducing space availability for shoppers and diners.

A recent survey of downtown shoppers indicated that over seventy percent of respondents cited fear of receiving a parking ticket as their main reason for cutting trips short.

Neighboring municipalities that recently increased their parking violation fines experienced a fifteen percent decrease in overall downtown vehicle traffic.

Sales tax rates in Elmridge are slightly higher than those in surrounding suburban commercial shopping centers.

A clear majority of downtown retail merchants currently favor the city council's proposed parking modification.

Show answer & explanation

Answer: Eliminating time limits causes downtown office commuters to occupy parking spaces early in the morning for full workdays, drastically reducing space availability for shoppers and diners.

Answer

The argument is most seriously weakened by the finding that removing parking time limits will cause full-day office commuters to occupy downtown parking spaces early in the morning, thereby reducing space availability for shoppers and diners throughout the day.

The correct answer weakens the argument by demonstrating an unintended side effect of the plan: full-day office commuters will park early and occupy downtown spaces all day. This severely reduces parking availability for retail customers and diners, neutralizing the intended revenue increase.

Step-by-Step Solution

1

Deconstruct the core argument

Premise: Eliminating the 2-hour parking limit allows drivers to park indefinitely. Conclusion: Visitors will stay longer to shop/dine, raising local business sales and city sales tax revenues.

Identifying the link between the plan (removing time limits) and the intended outcome (increased shopper visits and revenue) is essential to finding the unstated assumption.

2

Identify the unstated assumption

The argument assumes that shoppers and diners will be the primary beneficiaries of open-ended parking and will find available spaces easily.

A plan-to-goal argument relies on the assumption that the implementation will operate as intended without creating obstacles that defeat the goal.

3

Evaluate the impact of new evidence

If commuters occupy the spaces all day before shoppers arrive, overall customer turnover drops and shoppers cannot park downtown.

Introducing an alternative user group (commuters monopolizing spaces) invalidates the key assumption and demonstrates that the plan will decrease shopper access, directly undermining the conclusion.

Key Concept

Weakening Arguments - Plan Implementation and Unintended Consequences

Question 11Question →

A long-term agricultural study in a semi-arid basin established that whenever seasonal rainfall drops below 250 millimeters, farmers who irrigate their crops using unlined soil canals lose at least 35 percent of their diverted water to seepage. Conversely, any farm in the basin that adopts high-efficiency drip irrigation retains at least 85 percent of its diverted water regardless of seasonal rainfall levels. Last season, seasonal rainfall in the basin was 210 millimeters, yet Farm X lost less than 20 percent of its diverted water.

If the statements above are true, which of the following MUST also be true regarding Farm X last season?

Farm X did not rely on unlined soil canals to irrigate its crops last season.

Farm X used high-efficiency drip irrigation to irrigate its crops last season.

If Farm X had received more than 250 millimeters of rainfall last season, its diverted water loss would have been greater than 35 percent.

Farm X diverted a greater total volume of water last season than did neighboring farms using unlined soil canals.

Any farm in the basin that loses less than 35 percent of its diverted water during a low-rainfall season must be using high-efficiency drip irrigation.

Show answer & explanation

Answer: Farm X did not rely on unlined soil canals to irrigate its crops last season.

Answer

Farm X did not rely on unlined soil canals to irrigate its crops last season.

The correct answer is derived using formal conditional logic. The stimulus establishes that when rainfall is under 250 mm, using unlined soil canals guarantees a water loss of at least 35%. Since the rainfall was 210 mm and Farm X lost less than 20% of its diverted water, Farm X cannot have used unlined soil canals. By contrapositive reasoning, if the outcome (loss ≥ 35%) did not occur despite the trigger condition (rainfall < 250 mm) being met, the requirement for that outcome (unlined soil canals) must have been absent.

Step-by-Step Solution

1

Analyze the conditional rule for unlined soil canals.

If (Rainfall < 250 mm) AND (Unlined Soil Canals), then (Water Loss ≥ 35%).

This is a direct premise established by the study.

2

Evaluate the conditions given for Farm X last season.

Rainfall was 210 mm (which is strictly less than 250 mm), and Farm X's Water Loss was strictly less than 20%.

These are the explicit facts provided about Farm X.

3

Apply the contrapositive of the conditional rule.

Since Water Loss was NOT ≥ 35% (it was < 20%) while Rainfall was < 250 mm, the condition 'Unlined Soil Canals' must be FALSE for Farm X.

If P and Q imply R, then P and not-R implies not-Q.

Key Concept

Applying Contrapositive Logic in Must-Be-True Deductions

Question 12Question →

[Tab 1: Municipal Resilience Grant Policy (2026 Directive)]

Regional Infrastructure Authority Guidelines:
- Standard Eligibility: A municipal project is eligible for a base grant equal to 40% of its projected cost if it has a Resilience Index of at least 70 (out of 100) AND serves a target population of at least 50,000 residents.
- High-Impact Bonus: Projects that meet standard eligibility AND serve a target population exceeding 100,000 residents receive an additional 10% bonus grant (total grant of 50% of projected cost).
- High Flood Risk Exception: Any project located in a designated High Flood Risk zone has its Resilience Index requirement reduced to a minimum of 60. However, the maximum total grant funding awarded to any single project under this exception is strictly capped at $2,000,000 regardless of percentage calculations.

[Tab 2: Fiscal Year 2026 Project Applications]

Project ID	Project Type	Projected Cost	Resilience Index	Target Population	Location Zone
Project Alpha	Sea Wall Construction	$6,000,000	65	120,000	Zone R2
Project Beta	Grid Storage Backup	$4,500,000	72	80,000	Zone R1
Project Gamma	Urban Drainage System	$5,000,000	62	60,000	Zone R3
Project Delta	Bridge Structural Retrofit	$3,000,000	75	45,000	Zone R1

[Tab 3: Environmental Audit Memorandum]

Environmental Protection Board Assessment:
- Zone R1: Classified as Low Flood Risk.
- Zone R2: Classified as High Flood Risk due to coastal storm surge exposure.
- Zone R3: Classified as Moderate Flood Risk.
- Special Variance Note: No population threshold exceptions or policy variances were approved for any project in the 2026 funding cycle.

Based on the policy guidelines, project application data, and environmental audit memorandum, what is the total dollar amount of grant funding awarded across all four municipal project applications?

$1,800,000

$3,800,000

$4,800,000

$5,000,000

$6,800,000

Show answer & explanation

Answer: $3,800,000

Answer

$3,800,000

Synthesizing data across all three tabs demonstrates that Project Alpha qualifies under the High Flood Risk exception (Zone R2 per Tab 3) for a capped grant of

2,000,000. Project Beta qualifies under standard policy (Zone R1 per Tab 3) for a grant of 40% of

4,500,000 =

1,800,000. Projects Gamma and Delta fail eligibility criteria. Total funding awarded equals

3,800,000.

Step-by-Step Solution

1

Evaluate Project Alpha eligibility and grant amount by synthesizing data across Tab 1, Tab 2, and Tab 3.

Project Alpha is in Zone R2 (High Flood Risk per Tab 3). Tab 1 lowers the required Resilience Index from 70 to 60 for High Flood Risk zones. Project Alpha's Resilience Index of 65 satisfies this reduced threshold. Its population of 120,000 (>100,000) would qualify for a 50% grant (

3,000,000), but Tab 1 mandates a strict cap of

2,000,000 for High Flood Risk exception grants. Thus, Project Alpha receives $2,000,000.

Applying the conditional exception and maximum cap rules specified in Tab 1.

2

Evaluate Project Beta eligibility and grant amount.

Project Beta is in Zone R1 (Low Flood Risk per Tab 3). Standard policy applies. Its Resilience Index of 72 (>=70) and target population of 80,000 (>=50,000) meet standard criteria. It receives 40% of its

4,500,000 cost =

1,800,000.

Standard eligibility criteria check for non-high-risk projects.

3

Evaluate Project Gamma and Project Delta eligibility.

Project Gamma (Zone R3, Moderate Flood Risk) has a Resilience Index of 62, which is below the standard threshold of 70 (the reduced 60 threshold applies ONLY to High Flood Risk zones). Thus, Project Gamma receives

0. Project Delta has a population of 45,000, which is below the minimum threshold of 50,000 (and Tab 3 confirms no variances were approved). Thus, Project Delta receives

0.

Checking standard rule compliance and boundary conditions.

4

Calculate the total grant funding awarded.

2,000,000 (Project Alpha) +

1,800,000 (Project Beta) = $3,800,000.

Summing approved funding across all four applications.

Key Concept

Multi-Source Reasoning requiring cross-tab synthesis of policy rules, quantitative tables, and environmental risk classifications.

Estimated Time:2m 30s

Question 13Question →

If $x$ and $y$ are positive integers, is $x^3 - y^3$ divisible by $3$ ?

(1) $x + 2y$ is divisible by $3$ .
(2) $x^2 - y^2$ is divisible by $3$ .

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

Answer: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

The question asks whether

x^3 - y^3

is divisible by 3. By Fermat's Little Theorem or factoring

a^3 - a = a(a-1)(a+1)

(the product of three consecutive integers, always divisible by 3), any integer cubed has the same remainder when divided by 3 as the integer itself (

a^3 \equiv a \pmod 3

). Therefore,

x^3 - y^3

is a multiple of 3 if and only if

x - y

is a multiple of 3. Statement (1) specifies that

x + 2y

is divisible by 3. We can rewrite

x + 2y

as

(x - y) + 3y

. Because

3y

is automatically a multiple of 3,

(x - y)

must also be divisible by 3. This yields a definitive 'Yes', so Statement (1) alone is sufficient. Statement (2) states that

x^2 - y^2 = (x - y)(x + y)

is divisible by 3. Because 3 is prime, this requires 3 to divide either

(x - y)

or

(x + y)

. If

x = 4, y = 1

, 3 divides

(x - y) = 3

, giving a 'Yes'. If

x = 2, y = 1

, 3 divides

(x + y) = 3

while

(x - y) = 1

, giving

x^3 - y^3 = 7

, which is a 'No'. Hence, Statement (2) alone is not sufficient.

Step-by-Step Solution

1

Rephrase the question stem using algebraic modular properties.

Since

a^3 \equiv a \pmod 3

for any integer

a

,

x^3 - y^3 \equiv x - y \pmod 3

. Thus,

x^3 - y^3

is divisible by 3 if and only if

x - y

is divisible by 3.

Simplifying the target expression reduces the problem to determining whether 3 divides

(x - y)

.

2

Evaluate Statement (1):

x + 2y

is divisible by 3.

x + 2y = (x - y) + 3y

. Since

3y

is always a multiple of 3,

(x + 2y)

is divisible by 3 if and only if

(x - y)

is divisible by 3.

This guarantees a definitive 'Yes' to the target question. Statement (1) alone is SUFFICIENT.

3

Evaluate Statement (2):

x^2 - y^2

is divisible by 3.

x^2 - y^2 = (x - y)(x + y)

. Since 3 is prime, 3 must divide

(x - y)

or

(x + y)

. If

x = 4, y = 1

, then

x^2 - y^2 = 15

(divisible by 3) and

x^3 - y^3 = 63

(divisible by 3) -> YES. If

x = 2, y = 1

, then

x^2 - y^2 = 3

(divisible by 3), but

x^3 - y^3 = 7

(NOT divisible by 3) -> NO.

Statement (2) yields both 'Yes' and 'No' cases. Statement (2) alone is NOT SUFFICIENT.

Key Concept

Divisibility and Modular Congruences in Integer Properties

Estimated Time:2m 0s

Question 14Question →

To decrease total operating expenses, the Fairview Municipal Water District plans to install bio-purification filters upstream of its primary membrane filtration system. Water district officials note that microplastic debris currently clogs and damages the delicate membrane filters, requiring $2 million annually in replacement and repair costs. Because the new bio-purification filters effectively capture over 90 percent of incoming microplastics before they reach the primary membranes, officials conclude that installing these bio-filters will significantly reduce the facility's overall annual operating costs.

Which of the following, if true, most seriously weakens the water district officials' argument?

The primary membrane filters deteriorate twice as fast when processing water that contains high levels of microplastic debris compared to water free of microplastics.

Several neighboring water districts have successfully reduced their energy consumption by switching to alternate energy sources for their pumping stations.

The chemical solutions required to routinely clean and maintain the bio-purification filters cost substantially more per year than the total annual expenditure for replacing damaged membrane filters.

The microplastics currently captured by water treatment facilities pose no known health risks to consumers at current concentrations.

Although bio-purification filters take several months to manufacture, the facility's existing membrane filters can remain operational until the installation is complete.

Show answer & explanation

Answer: The chemical solutions required to routinely clean and maintain the bio-purification filters cost substantially more per year than the total annual expenditure for replacing damaged membrane filters.

Answer

The chemical solutions required to routinely clean and maintain the bio-purification filters cost substantially more per year than the total annual expenditure for replacing damaged membrane filters.

The conclusion asserts that adding bio-purification filters will reduce overall annual operating expenses by eliminating

2 million in membrane repairs. This logic relies on the assumption that running the bio-filters will not generate new expenses equal to or greater than

2 million. The correct response points out that routine maintenance chemicals for the bio-filters cost substantially more per year than the $2 million saved, demonstrating that the net financial outcome will be an increase in total operating expenses, directly weakening the conclusion.

Step-by-Step Solution

1

Deconstruct the argument structure

Premise: Microplastics cause $2 million in annual damage to membrane filters. Premise: Bio-filters remove 90% of microplastics before reaching membranes. Conclusion: Installing bio-filters will reduce overall annual operating costs.

Identifying the central claim and supporting evidence reveals the underlying assumption.

2

Identify the unstated assumption

The argument assumes that the ongoing cost to purchase, operate, and maintain the new bio-filters will be significantly less than the $2 million currently spent on membrane repairs.

A net reduction in total operating costs requires that savings exceed any new expenses incurred by the plan.

3

Evaluate choices for an undermining fact

The statement showing that chemical cleaning solutions for the bio-filters cost more per year than the $2 million membrane repair bill directly breaks the underlying assumption.

If new operational expenses exceed current repair costs, total operating expenses will increase rather than decrease.

Key Concept

Evaluating Plan Feasibility and Net Cost-Benefit Assumptions

Question 15Question →

To reduce regional water consumption, an agricultural cooperative plans to distribute automated soil-moisture sensors to all member farms. The sensors send real-time alerts whenever soil moisture drops below an optimal threshold, allowing farmers to irrigate only when necessary rather than on a fixed calendar schedule. The cooperative's directors conclude that installing these sensors will significantly decrease total agricultural water usage across the region over the next two years. Which of the following, if true, most seriously weakens the directors' argument?

Fixed-schedule irrigation routinely applies water during afternoon hours of peak evaporation, whereas sensor-based watering typically occurs during cooler evening hours.

The manufacturing process for solar-powered moisture sensors produces industrial chemical waste that requires specialized recycling facilities.

Most member farmers currently irrigate less than the optimal moisture threshold due to water cost concerns, but plan to increase watering to recommended sensor levels once alerts are automated.

Neighboring agricultural districts that installed similar moisture sensors three years ago experienced a minor increase in local technology maintenance jobs.

The automated moisture sensors require periodic recalibration if exposed to extreme frost during winter months.

Show answer & explanation

Answer: Most member farmers currently irrigate less than the optimal moisture threshold due to water cost concerns, but plan to increase watering to recommended sensor levels once alerts are automated.

Answer

The directors' argument is most seriously weakened by the finding that most member farmers currently underwater their crops relative to the optimal threshold and intend to increase watering once sensor alerts are implemented.

The conclusion relies on the unstated assumption that current schedule-based irrigation uses more water than sensor-guided optimal irrigation. The correct choice reveals that farmers currently apply less water than the optimal threshold due to costs, and will increase watering once sensor feedback is available. This counter-behavior causes total water consumption to rise, directly undermining the cooperative's objective.

Step-by-Step Solution

1

Identify the argument's premises and conclusion.

Premise: Automated sensors alert farmers to irrigate only when moisture drops below an optimal threshold, replacing fixed schedules. Conclusion: Regional agricultural water usage will decrease significantly.

Understanding the logical jump from premise to conclusion reveals the implicit assumption.

2

Uncover the central unstated assumption.

The argument assumes that farmers' current fixed schedules deliver more water than the sensor-determined optimal threshold requires.

If current watering is equal to or less than optimal levels, adopting precision sensors will not reduce water volume.

3

Evaluate the choices to find new evidence breaking this assumption.

The statement showing that farmers currently underwater their crops and plan to raise irrigation to match sensor recommendations directly refutes the assumption of net water savings.

If farmers increase watering frequency upon receiving sensor data, overall regional consumption will rise rather than fall.

Key Concept

Weakening Plan-to-Goal Arguments by Identifying Unintended Behavioral Counter-Effects

Question 16Question →

A wooden box contains 15 identical tokens, each marked with a distinct integer from $1$ to $15$ , inclusive. If one token is drawn at random from the box, what is the probability that the integer on the drawn token is a prime number?

\frac{1}{3}

\frac{2}{5}

\frac{7}{15}

\frac{8}{15}

\frac{2}{3}

Show answer & explanation

Answer: $\frac{2}{5}$

Answer

The probability that the integer on the drawn token is a prime number is

\frac{2}{5}

.

The total number of possible outcomes when choosing one token from

15

tokens is

15

. The prime numbers between

1

and

15

, inclusive, are

2, 3, 5, 7, 11,

and

13

. There are

6

favorable outcomes. The probability is therefore

\frac{6}{15}

, which simplifies to

\frac{2}{5}

.

Step-by-Step Solution

1

Determine the total number of possible outcomes (the denominator).

The total number of tokens is

15

, so

N = 15

.

The sample space consists of all integers from

1

to

15

, inclusive.

2

Identify and count all prime numbers in the set from

1

to

15

, inclusive.

The prime numbers in this set are

2, 3, 5, 7, 11,

and

13

. Thus, there are

6

prime numbers.

A prime number is an integer strictly greater than

1

that has exactly two distinct positive divisors:

1

and itself. Note that

1

is not prime.

3

Calculate the basic probability

P(\text{Prime}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}

.

P(\text{Prime}) = \frac{6}{15} = \frac{2}{5}

.

Dividing the favorable count

6

by total count

15

and simplifying by dividing numerator and denominator by

3

yields

\frac{2}{5}

.

Key Concept

Basic Single-Event Probability and Definition of Prime Numbers

Estimated Time:45s

Question 17Question →

If $r$ and $s$ are the two distinct real roots of the quadratic equation $x^2 - 6x + 4 = 0$ , what is the value of $\frac{r^3 + s^3}{r + s}$ ?

24

28

32

36

44

Show answer & explanation

Answer: $24$

Answer

The value of

\frac{r^3 + s^3}{r + s}

is

24

.

Using Vieta's formulas for

x^2 - 6x + 4 = 0

, the sum of the roots is

r + s = 6

and the product is

rs = 4

. Factoring

r^3 + s^3

gives

(r + s)(r^2 - rs + s^2)

. Dividing by

(r + s)

leaves

r^2 - rs + s^2

, which can be rewritten as

(r + s)^2 - 3rs

. Substituting the Vieta values yields

6^2 - 3(4) = 36 - 12 = 24

.

Step-by-Step Solution

1

Apply Vieta's formulas to the given quadratic equation

x^2 - 6x + 4 = 0

.

The sum of the roots is

r + s = 6

, and the product of the roots is

rs = 4

.

For a quadratic equation

x^2 + bx + c = 0

, Vieta's formulas state that the sum of roots is

-b

and the product of roots is

c

.

2

Factor the sum of cubes expression

r^3 + s^3

.

\frac{r^3 + s^3}{r + s} = \frac{(r + s)(r^2 - rs + s^2)}{r + s} = r^2 - rs + s^2

.

The sum of cubes factors algebraically into

(r + s)(r^2 - rs + s^2)

, and

r + s = 6 \neq 0

allows cancellation.

3

Express

r^2 - rs + s^2

in terms of

(r + s)

and

rs

.

r^2 - rs + s^2 = (r + s)^2 - 3rs

.

Since

(r + s)^2 = r^2 + 2rs + s^2

, subtracting

3rs

yields

r^2 - rs + s^2

.

4

Substitute the known values

r + s = 6

and

rs = 4

into the expression.

6^2 - 3(4) = 36 - 12 = 24

.

Evaluating the algebraic expression yields the final value.

Key Concept

Polynomial Factoring and Vieta's Formulas for Quadratic Equations

Estimated Time:1m 30s

Question 18Question →

Determine the sum of all real solutions to the equation $|x^2 - 4x| = 3x - 6$ .

7

8

9

10

12

Show answer & explanation

Answer: $9$

Answer

The sum of all valid real solutions is

9

.

The correct answer is

9

. Setting up the two cases

x^2 - 4x = 3x - 6

and

x^2 - 4x = -(3x - 6)

yields candidate roots

x = 1, 6, 3,

and

-2

. Because the absolute value expression

|x^2 - 4x|

cannot be negative,

3x - 6

must be non-negative, requiring

x \ge 2

. Evaluating each candidate shows that

x = 1

and

x = -2

produce negative right-hand sides and are extraneous. The only valid solutions are

x = 3

and

x = 6

, whose sum is

3 + 6 = 9

.

Step-by-Step Solution

1

Establish the domain condition for the right-hand side of the absolute value equation.

Since absolute values are non-negative,

|x^2 - 4x| \geq 0

requires

3x - 6 \geq 0 \implies x \geq 2

.

An absolute value expression cannot equal a negative number.

2

Solve Case 1 where

x^2 - 4x = 3x - 6

.

x^2 - 7x + 6 = 0 \implies (x - 1)(x - 6) = 0 \implies x = 1

or

x = 6

.

This corresponds to the positive branch of the absolute value.

3

Solve Case 2 where

x^2 - 4x = -(3x - 6)

.

x^2 - 4x = -3x + 6 \implies x^2 - x - 6 = 0 \implies (x - 3)(x + 2) = 0 \implies x = 3

or

x = -2

.

This corresponds to the negative branch of the absolute value.

4

Test all candidate solutions (

x = -2, 1, 3, 6

) against the domain constraint

x \geq 2

.

x = -2

yields

3(-2)-6 = -12 < 0

(extraneous).

x = 1

yields

3(1)-6 = -3 < 0

(extraneous).

x = 3

yields

|9-12| = 3 = 3(3)-6

(valid).

x = 6

yields

|36-24| = 12 = 3(6)-6

(valid).

Extraneous roots introduced by unconstrained case splitting must be eliminated.

5

Sum the valid real solutions.

3 + 6 = 9

.

The question asks specifically for the sum of all valid real solutions.

Key Concept

Absolute Value Equations and Extraneous Solution Verification

Question 19Question →

In a corporate evaluation of $200$ employees, performance scores range from $0$ to $100$ . An employee with a score of $84$ scored strictly higher than $150$ employees and strictly lower than $42$ employees, while the remaining employees received a score of exactly $84$ . If the percentile rank of a score is defined as the percentage of all scores strictly below it plus half the percentage of all scores equal to it, what is the percentile rank of a score of $84$ ?

75th percentile

77th percentile

79th percentile

84th percentile

88th percentile

Show answer & explanation

Answer: 77th percentile

Answer

77th percentile

The correct answer represents the relative position of a score of 84. Since 150 employees scored lower and 8 employees scored equal to 84, applying the percentile formula yields ((150 + 4) / 200) * 100 = 77th percentile.

Step-by-Step Solution

1

Determine the number of employees scoring exactly 84.

Number of employees scoring 84 =

200 - 150 - 42 = 8

.

The total group size equals the sum of students scoring below 84, above 84, and equal to 84.

2

Calculate the effective count of scores at or below 84 under the given percentile definition.

Effective count =

150 + 0.5 \times 8 = 154

.

The definition specifies adding the count of strictly lower scores to half the count of tied scores.

3

Convert the effective count to a percentile rank.

Percentile rank =

\frac{154}{200} \times 100 = 77\text{th percentile}

.

Dividing by the total number of scores (

200

) and multiplying by

100

converts the count into a percentage.

Key Concept

Percentile Rank Calculation

Estimated Time:1m 30s

Question 20Question →

In a survey of $120$ corporate executives, $70$ executives read Magazine A, $50$ read Magazine B, and $25$ read both Magazine A and Magazine B. How many of the executives surveyed read neither Magazine A nor Magazine B?

0

25

45

50

95

Show answer & explanation

Answer: 25

Answer

25 executives surveyed read neither Magazine A nor Magazine B.

By the inclusion-exclusion principle, the total number of executives reading at least one magazine is given by

\text{Set A} + \text{Set B} - \text{Both} = 70 + 50 - 25 = 95

. Since

120

executives were surveyed in total, the number of executives who read neither magazine is

120 - 95 = 25

.

Step-by-Step Solution

1

Apply the Inclusion-Exclusion Principle for two overlapping sets to find the number of executives reading at least one magazine.

\text{Total(At least one)} = \text{Magazine A} + \text{Magazine B} - \text{Both} = 70 + 50 - 25 = 95

.

Directly adding the counts for Magazine A and Magazine B counts executives who read both twice, so the intersection must be subtracted once.

2

Subtract the number of executives who read at least one magazine from the total number of executives surveyed.

\text{Neither} = \text{Total Surveyed} - \text{Total(At least one)} = 120 - 95 = 25

.

The universe of surveyed executives consists of those who read at least one magazine and those who read neither.

Key Concept

Overlapping Sets (Double-Matrix / Two-Set Venn Diagram)

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

Answer

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Key Concept